Mathematics > Operator Algebras

Title:
Cohomology for small categories: $k$-graphs and groupoids

Abstract: Given a higher-rank graph $\Lambda$, we investigate the relationship between
the cohomology of $\Lambda$ and the cohomology of the associated groupoid
$G_\Lambda$. We define an exact functor between the abelian category of right
modules over a higher-rank graph $\Lambda$ and the category of
$G_\Lambda$-sheaves, where $G_\Lambda$ is the path groupoid of $\Lambda$. We
use this functor to construct compatible homomorphisms from both the cohomology
of $\Lambda$ with coefficients in a right $\Lambda$-module, and the continuous
cocycle cohomology of $G_\Lambda$ with values in the corresponding
$G_\Lambda$-sheaf, into the sheaf cohomology of $G_\Lambda$.

Comments:

A flaw in the proof of Proposition 4.2 in v1 of this paper has invalidated Proposition 4.8 and Theorem 4.9 from v1. This version (v3) has been substantially revised and includes new results. Version 4 to appear in Banach J. Math. Anal